Despite all the Tebow-madness, there were quite a few other games decided in the waning seconds. With 0:26 left in the game, Joe Webb and the Vikings were faced with a 4th-and-6 from the Detroit 12, down 6. If the Vikings were only down 3, this is a no brainer: Kicking the field goal results in a win probability of 43% versus only 33% if they go for it. Detroit, however, had tacked on a field goal early in the 4th quarter and their defense came up huge, stopping Webb at the goal line (ignoring any questionably missed face mask calls on the last play of the game). Let's take a look at Detroits' last scoring drive.

With 4:21 left in the 3rd, Matthew Stafford led a 7-minute procession that began on his own 10-yardline. The Lions converted on three separate 3rd-downs to keep the drive alive, the biggest of which was a 25-yard completion to Titus Young on 3rd-and-1 from their own 19. Using our Markov model, we can see how the drive developed:

The two biggest conversions were the aforementioned Young completion, which chopped the 63% punt probability in half, and a 7-yard 3rd-and-6 completion (play 9) to Brandon Pettigrew. Before the Pettigrew conversion, there was a 46% chance that the drive would end in a punt; after the catch, that probability dropped to 10% and the scoring probability jumped to 61%.

The Lions were ultimately stopped on a 3rd-and-2, took a delay of game penalty to make it 4th-and-6 and kicked the field goal on 4th down. Using Brian's nifty 4th Down Calculator, we can analyze Detroit's decision to kick the field goal. In terms of efficiency, going for it on the 4th-and-1, before the delay of game penalty, converts at 68% giving a total of 4.21 expected points. A field goal converts very close to 100%, yielding 2.39 expected points. Since it was late in the game, win probability is equally important in making the decision. Both going for it and kicking the field goal result in a 97% win probability; the break even point in terms of making the decision is a conversion rate of 66%.

Given a late surge by the Vikings, the field goal ended up making a substantial difference - although a 4th-down touchdown conversion would have made it a three-score game. Because of the length of the drive, the Lions' win probability increased by 14% which proved to be substantial.

So, the question becomes, in a situation where you are almost assured victory, do you take the guaranteed 3 points or follow expectation? Since the decision has a negligible affect on your team's chances of winning the game, a decision-maker cannot really be faulted one way or the other - even if the math leans ever-so-slightly toward going for it.

I'll bite. Why does he need to incorporate intangibles? He's already told you how Denver is winning - defense and special teams.

Did you not notice that Tebow was off the field for the four biggest plays of the game, the two 50+ yds field goals and Marion Barbers two fails? Even if I believed in 'the ability to win' I just don't see how even Tebow can project that ability onto the field when he's on the sideline.

And if Prater missed the 59 YARD FIELD GOAL(!), then it looks like Tebow CHOKED! He had 3 downs and couldn't gain a yard. Right? Also if Tebow's offense manages to score more than 10 points in regulation, the Broncos win the game easily. If the defense lets up, say, 17 points, Tebow sucks and isn't a NFL QB. People's analysis of Tebow is so tied up with things completely unrelated to him. At this point, the best approach might be to just ignore all Tebow talk.

Although the math says there's only a slight advantage to going for it on the 4th down at the end of the Lions' drive, I think in practice the benefit is huge. This blog chronicles the overwhelming conservatism of most NFL coaches, and that is what tips the scales for me.

If the conversion succeeds, the game is essentially over. If it fails, the Vikings can play conservative because they just need a field goal to tie. But if the Lions kick a field goal, then the Vikings are forced to pull out all the stops and go for a touchdown. By kicking a field goal, the Lions forced their opponent to make more aggressive decisions, which (as detailed elsewhere in this blog) tend to yield higher results.

For me, the conservative mentality of most coaches in the NFL means that the LIons maximized their chance to lose by kicking the field goal. If you go for it and convert you win; if you fail the opponent plays for the tie. Only by going up six do you leave them no other choice but to play aggressive.

That's a good point. It's been hypothesized elsewhere on this site that putting a team down by 6 can theoretically be more dangerous than putting them down by 3. Look at Super Bowl XLV - the Packers kicked a field goal to go up by 6, thus giving the ball back to the Steelers who were focused entirely on winning with a TD rather than possibly winning but maybe just tying with a FG. A team in that position can plan to go for it on 4th down no matter the situation, but if they're down by 3 they'll usually start phoning it in once they get past the opponent's 30 yard line.

Every week I hear a different excuse. One week it's if the defense didn't make this tackle. The next week is if the kicker didn't make this kick. Next week it will be if the OC didn't call this play. The only staple in the 4th quarter the last 7 weeks is Tebow's magic (AKA intangibles). MVP!

Tebow and only Tebow. Do you have any idea what Tebow would have done to Prater if he missed those FGs?

"3. Barber runs out of bounds in the 4th qtr and then fumbles on the DEN 33 in OT".

Yup. Tebow and Tebow. He enforced his willpower on poor Barber.

I'm with Burke. Broncos are winning not because of Tebow, just like Niners are winning not because of Smith. Defense, special teams, coaching, game plans, lot of luck. We, people, are bad analysing randomness.

Yeah, interesting point. I definitely agree that by forcing the Vikings to go for a field goal it essentially improves their decision-making due to the conservative nature of NFL coaches.

I wonder if there has been a discussion about utility in the mindset of decision-makers. That is, when we typically talk about the math involved, our only goal is to win the football game. In reality, however, NFL coaches have many other things to deal with - like keeping their job and maintaining a position of respect in the club house. Since the media tends to crucify risky decisions that don't pay off, coaches are incentivized to play conservative strategies. Their is a balance of utility between actually winning games and being seen as a competent coach. It could be possible - although I doubt it due to the degree of the market inefficiencies - that coaches are actually maximizing their utility by staying conservative.

And that is the idea that the media does not equally reward coaches for making risky decisions that DO payoff. The punishment for the risk paying off tends to be much more severe (we don't hear about the good plays).

I know someone else mentioned a school assignment about Tebow last week.Something similar happened in my kids writing class today. A student wrote an essay titled "Is Tim Tebow Jesus Christ?". I thought he would get suspended. Apparently he got an A plus.

I think prospect theory probably does explain a lot of coaches' conservatism, but your suggestion that coaches are factoring in their job security (also probably correct) brings up an interesting way to prove or disprove this hypothesis.

If coaches do factor in job security in their decision making, coaches who are highly unlikely to be fired in the near future should make better decisions on (for instance) fourth down. Comparing the fourth down decision making of coaches who have made the playoffs in consecutive years (or who are just coming off a deep playoff run) to that of coaches who are struggling might illustrate how significant this effect is. The trouble would be sorting out the selection bias -- maybe the coaches that have made the playoffs for consecutive years were already making closer to optimal decisions.

Maybe a way to account for selection bias would be to do the same analysis with coaches who were about to be fired -- if anything, they'd know that they're about to be axed and would play less conservatively in an attempt to win (and save their jobs). This assumption could be wrong (maybe they play conservatively so that less of the blame for the losses is attributed to them), but the worst an analysis can be is inconclusive.

I think the title of the graph should be changed to "Lions' 4th Quarter Field Goal Drive" if possible. I spent a minute wondering how a drive that started in the 3rd quarter could have a 10% probability of finishing with the end of the game.

Detroit's third-down and short-yardage offense has been significantly below average this season. They don't have a true fullback (they use TEs when they go to two-back sets) and were missing the only two backs who've been effective this season (Best and Smith). With that in mind, I'm not surprised that Schwartz and Linehan elected to kick.

Anecdotally, Stafford has made a number of poor decisions in the second half of the season, so while the prospect of a short pass to Pettigrew or Johnson might have been enticing, the possibility of giving the Vikings the ball at the 20 (or worse, depending on the return) may have outweighed it.

Keith, I am curious - you note the relative chances of each strategy succeeding and the expected EP of each. But EP is an average-based statistic, correct? Do you guys ever worry about the distribution (variance/error) of outcomes and risk-considerations when assessing decisions? Ie, it's true that on average the go-for-it decisions pays off more, but it doesn't seem like you're always concerned about the average, particularly if the error is not actually normally distributed around that outcome.

Parallel example to help exemplify what I'm thinking about here - you have $100 to your name, and you can either invest it in stock A which will come back as $110 100% of the time or stock B which will come back as $0 99% of the time and $100,100 1% of the time. The average (expected) payoff of stock A is $10, and the expected payoff of B is $100,000x.01-100*.99 = $901. So B has a MUCH higher EP, but it carries a huge risk of losing all of your money...

That example is exaggerated, of course, but is there a parallel here? Going for it would yield more payoff in the aggregate if you do this over and over again, but what about *this* decision - is there a way to factor in the risk of failure that doesn't just look at averages?

Also, the point about inadvertently optimizing your opponent's behavior by going up 6 instead of staying at 3 is both interesting and hilarious. If that had a huge effect, I imagine you would forget about the field goal entirely.

Tobias Moskowitz has looked at this in his recent book "Scorecasting: The Hidden Influences Behind How Sports Are Played and Games Are Won."

He finds that coaches do indeed become more appropriately aggressive when they gain job security. As an example, he mentioned Bill Belichik, who now is one of the most aggressive coaches on 4th down. When he was a younger coach and less secure in his job, he was not nearly as aggressive on 4th down.

I think the Lions game, as well as the Saints-Titans game, illustrate this point:Yes, going ahead or being ahead by 4+ points (but less than 8) "makes" your opponent go for the TD--but your D can play differently, knowing that the opponent has to pick up 30+ more yds. On the last play of both games, MIN & TEN had to pass up "chip-shot" FG's--which they would have surely taken in the same situation at the end of the FIRST half--to attempt to score the game-winning TD. There is value in the offense needing those extra yards to tie/win.BTW, we understand that the trailing offense plays more aggressively to score the TD--but they also must use a higher-risk strategy. Since they HAVE TO gain those extra 30+ yds, they attempt riskier downfield passes--which are more likely to be incomplete/intercepted. IMO, HC's are saying, "If you are going to have a chance to beat me, I will make you take higher risks, and you will have to score a TD. I am going to make it as hard on you as possible." Let's face it, if the defense isn't able in that instance to stop them from scoring a last-minute TD, they weren't keeping them out of FG range, and may not have kept them out of close FG range. Sure, the trailing OC may play a more conservative strategy, or the K may miss it, but the winning HC is making the other team play riskier than they would choose to if there were the whole 4th Q to play.

Dan, to your point, that's one of the things that baffles me about Jason Garrett. Jerry Jones specifically stated earlier in the year the Cowboys should be more aggressive in late game situations, yet Garrett is still horribly conservative. He doesn't have the job security, but he does have the support to take those risks. So why not take them?! Does he not trust aggression helps, or is he not convinced by Jones's comments?

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